{"title":"平方和和多项式微积分中的单项大小与位复杂度","authors":"Tuomas Hakoniemi","doi":"10.1109/LICS52264.2021.9470545","DOIUrl":null,"url":null,"abstract":"In this paper we consider the relationship between monomial-size and bit-complexity in Sums-of-Squares (SOS) in Polynomial Calculus Resolution over rationals $({\\text{PCR}}/\\mathbb{Q})$. We show that there is a set of polynomial constraints Qn over Boolean variables that has both SOS and ${\\text{PCR}}/\\mathbb{Q}$ refutations of degree 2 and thus with only polynomially many monomials, but for which any SOS or ${\\text{PCR}}/\\mathbb{Q}$ refutation must have exponential bit-complexity, when the rational coefficients are represented with their reduced fractions written in binary.","PeriodicalId":174663,"journal":{"name":"2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"54 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Monomial size vs. Bit-complexity in Sums-of-Squares and Polynomial Calculus\",\"authors\":\"Tuomas Hakoniemi\",\"doi\":\"10.1109/LICS52264.2021.9470545\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we consider the relationship between monomial-size and bit-complexity in Sums-of-Squares (SOS) in Polynomial Calculus Resolution over rationals $({\\\\text{PCR}}/\\\\mathbb{Q})$. We show that there is a set of polynomial constraints Qn over Boolean variables that has both SOS and ${\\\\text{PCR}}/\\\\mathbb{Q}$ refutations of degree 2 and thus with only polynomially many monomials, but for which any SOS or ${\\\\text{PCR}}/\\\\mathbb{Q}$ refutation must have exponential bit-complexity, when the rational coefficients are represented with their reduced fractions written in binary.\",\"PeriodicalId\":174663,\"journal\":{\"name\":\"2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)\",\"volume\":\"54 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/LICS52264.2021.9470545\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/LICS52264.2021.9470545","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Monomial size vs. Bit-complexity in Sums-of-Squares and Polynomial Calculus
In this paper we consider the relationship between monomial-size and bit-complexity in Sums-of-Squares (SOS) in Polynomial Calculus Resolution over rationals $({\text{PCR}}/\mathbb{Q})$. We show that there is a set of polynomial constraints Qn over Boolean variables that has both SOS and ${\text{PCR}}/\mathbb{Q}$ refutations of degree 2 and thus with only polynomially many monomials, but for which any SOS or ${\text{PCR}}/\mathbb{Q}$ refutation must have exponential bit-complexity, when the rational coefficients are represented with their reduced fractions written in binary.